Monitoring focus of a lens imaging system based on astigmatism

A method for monitoring focus of a lens system, comprising:

• measuring imaged linewidths of a first line feature along a first orientation in an imaging plane of a lens system at a plurality of focus positions;
• measuring imaged linewidths of a second line feature of a design linewidth identical to a design linewidth of the first line feature, the second line feature being along a second orientation orthogonal to the first orientation in the imaging plane at the plurality of focus positions;
• subtracting imaged linewidths that are measured along the first and second orientations to obtain differences in linewidth at the plurality of focus positions;
• forming a monotonic function of the differences in linewidth with respect to focus positions;
• measuring a difference between linewidths of two orthogonal line features of the same linewidth in an image produced by the lens system;
• and
• determining a respective focus position of the image according to the monotonic function and determining a position difference between the respective focus position and a desired focus position.

A method of determining flatness of a wafer, comprising:

• calibrating a lens system that exhibits astigmatism to determine a monotonic functional dependence of differences in linewidth of two orthogonal line features of the same linewidth with respect to focus positions;
• placing a surface of a wafer at a known focus position to receive an image produced by the lens system;
• measuring a difference between linewidths of two line features of the same linewidth in the image on the surface;
• using the monotonic functional dependence of differences to determine a measured focus position;
• and
• determining a difference between the measured focus position and the known focus position to indicate the flatness of the surface.